Crosscap numbers of alternating links via state codes

TL;DR

This paper introduces a novel encoding method for Kauffman states to compute the crosscap number and genus of prime alternating links, providing new data and patterns for links and knots with up to 19 crossings.

Contribution

It presents a new encoding technique for Kauffman states and a procedure to determine crosscap numbers, expanding the computational understanding of prime alternating links and knots.

Findings

Computed crosscap numbers for all prime alternating links with up to 14 crossings.

Computed crosscap numbers for all prime alternating knots with up to 19 crossings.

Identified intriguing patterns in the computed data.

Abstract

We describe a way of encoding a Kauffman state as a set of tuples, similar to a Gauss code. Then we describe a procedure for using these state codes to determine the unoriented genus and crosscap number of any prime alternating knot or link. Finally, we compute these values for all such links through 14 crossings and all such knots through 19 crossings (this data is new for links with 10-14 crossings and knots with 14-19 crossings), and we identify several intriguing patterns in the resulting data.